Recall that \(G\)  has the Haagerup property if and only if there exists a sequence of normalized, positive definite functions \((\varphi _n)_{n\ge 1}\)  on \(G\)  such that \(\varphi _n\)  tends to the constant function 1 uniformly on compact subsets of \(G\)  as \(n \rightarrow \infty \) , and each \(\varphi _n \in C_0(G)\) . In turn, \(G\)  is amenable if and only if each \(\varphi _n\)  can be chosen with compact support. Hence the Haagerup property is a weaker property than amenability, which itself generalizes both abelian and finite groups. Even though, there exists groups with the Haagerup property that are not amenable, free groups for example, there are a lot of properties of amenable groups that still hold for groups with the Haagerup property.
For example, groups with the Haagerup property satisfy the Baum-Connes conjecture [1]}.
